Min. A most convenient abridgment.
Euc. Similarly, it admits of easy proof that, if a Pair of Lines make, with a certain transversal, either (a) a pair of alternate angles unequal, or (b) an exterior angle unequal to the interior opposite angle on the same side of the transversal, or (c) a pair of interior angles on the same side of the transversal not supplementary; they will make, with the same transversal, (d) each pair of alternate angles unequal, and (e) every exterior angle unequal to the interior opposite angle on the same side of the transversal, and (f) each pair of interior angles on the same side of the transversal not supplementary.
And when I speak of a Pair of Lines as 'unequally inclined to' a transversal, I wish it to be understood that they fulfil some one of the three conditions (a), (b), (c), and therefore all the three conditions (d), (e), (f).
Min. Very well.
Euc. Now the Propositions relating to Pairs of Lines may be divided into two classes, the first covering the ground occupied by my Axiom 10 ('two straight Lines cannot enclose a space') and my Propositions I. 16, 17, 27, 28, 31; the second that occupied by my Axiom 12 and Propositions I. 29, 30, 32. Those in the first class are logical deductions from Axioms which have never been disputed: the second class has furnished, through all ages, a battle-field for rival mathematicians. That some one of the Propositions in this class must be assumed as an Axiom is agreed on all hands, and each combatant in turn proclaims his own special favourite to be the one axiomatic truth of the series, insisting that all the rest ought to be proved as Theorems.
